C₂×#7676228 #72298745

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#7676228

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

120021 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 26 points:

(8,10)(15,23)(20,25)(21,26)(22,24);(3,4,5,6)(7,8)(9,10)(11,12,16,17)(13,18,19,14)(15,20,23,25)(21,24,26,22);(12,17)(14,18)(20,25)(22,24);(3,4,5,6)(11,13,16,19)(12,18,17,14)(15,21,23,26)(20,24,25,22);(3,4)(5,6)(11,14,16,18)(12,19,17,13)(15,24,23,22)(20,21,25,26);(11,15)(12,20)(13,21)(14,22)(16,23)(17,25)(18,24)(19,26);(1,2)

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297215
maximal normal subgroup of	C₂×#72297215
maximal quotient of	C₂×#72298075, C₂×#72298091

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((8,10)(15,23)(20,25)(21,26)(22,24), (3,4,5,6)(7,8)(9,10)(11,12,16,17)(13,18,19,14)(15,20,23,25)(21,24,26,22), (12,17)(14,18)(20,25)(22,24), (3,4,5,6)(11,13,16,19)(12,18,17,14)(15,21,23,26)(20,24,25,22), (3,4)(5,6)(11,14,16,18)(12,19,17,13)(15,24,23,22)(20,21,25,26), (11,15)(12,20)(13,21)(14,22)(16,23)(17,25)(18,24)(19,26), (1,2));

C2×#7676228 #72298745